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#### Article Title

The general $\phi$-Hermitian solution to mixed pairs of quaternion matrix Sylvester equations

#### Keywords

Quaternion, Sylvester-type equations, Moore-Penrose inverse, Involution, $\phi$-Hermitian solution, Rank

#### Abstract

Let $\mathbb{H}^{m\times n}$ be the space of $m\times n$ matrices over $\mathbb{H}$, where $\mathbb{H}$ is the real quaternion algebra. Let $A_{\phi}$ be the $n\times m$ matrix obtained by applying $\phi$ entrywise to the transposed matrix $A^{T}$, where $A\in\mathbb{H}^{m\times n}$ and $\phi$ is a nonstandard involution of $\mathbb{H}$. In this paper, some properties of the Moore-Penrose inverse of the quaternion matrix $A_{\phi}$ are given. Two systems of mixed pairs of quaternion matrix Sylvester equations $A_{1}X-YB_{1}=C_{1},~A_{2}Z-YB_{2}=C_{2}$ and $A_{1}X-YB_{1}=C_{1},~A_{2}Y-ZB_{2}=C_{2}$ are considered, where $Z$ is $\phi$-Hermitian. Some practical necessary and sufficient conditions for the existence of a solution $(X,Y,Z)$ to those systems in terms of the ranks and Moore-Penrose inverses of the given coefficient matrices are presented. Moreover, the general solutions to these systems are explicitly given when they are solvable. Some numerical examples are provided to illustrate the main results.

#### Recommended Citation

He, Zhuo-Heng; Liu, Jianzhen; and Tam, Tin-Yau.
(2017),
"The general $\phi$-Hermitian solution to mixed pairs of quaternion matrix Sylvester equations",
*Electronic Journal of Linear Algebra*,
Volume 32, pp. 475-499.

DOI: https://doi.org/10.13001/1081-3810.3606

*Abstract*